DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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$Math 664 Homework #2: Solutions 1. Let Î© = R, F = {A â R : either A ...$

4.3 Labelled Trees and Prüfer Codes . . . . . . . . . . . . . . . . 63 4.3.1 More information from Prüfer codes (Optional) . . . . 66 4.4 Monochromatic Subgraphs (Optional) . . . . . . . . . . . . . 66 4.5 Spanning Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.5.1 Counting the Number of Spanning Trees (Optional) . 70 4.6 Finding Shortest Paths in Graphs . . . . . . . . . . . . . . . . 72 4.7 Some Asymptotic Combinatorics (Optional) . . . . . . . . . . 73 5 Generating Functions 75 5.1 Using Pictures to Visualize Counting . . . . . . . . . . . . . . 75 5.1.1 Pictures of trees (Optional) . . . . . . . . . . . . . . . 78 5.2 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . 79 5.2.1 Generating polynomials . . . . . . . . . . . . . . . . . 79 5.2.2 Generating functions . . . . . . . . . . . . . . . . . . . 80 5.2.3 Product Principle for Generating Functions (Optional) 85 5.3 Solving Recurrences with Generating Functions . . . . . . . . 86 6 The Principle of Inclusion and Exclusion 89 6.1 The Size of a Union of Sets . . . . . . . . . . . . . . . . . . . 89 6.2 The Principle of Inclusion and Exclusion . . . . . . . . . . . . 91 6.3 Counting the Number of Onto Functions . . . . . . . . . . . . 93 6.4 The Ménage Problem . . . . . . . . . . . . . . . . . . . . . . 94 6.5 The Chromatic Polynomial of a Graph . . . . . . . . . . . . . 95 6.5.1 Deletion-Contraction (Optional) . . . . . . . . . . . . 96 7 Distribution Problems 98 7.1 The Idea of Distributions . . . . . . . . . . . . . . . . . . . . 98 7.2 Counting Partitions . . . . . . . . . . . . . . . . . . . . . . . 100 7.2.1 Multinomial Coefficients (Optional) . . . . . . . . . . 102 7.3 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . 102 II REVIEW MATERIAL 108 A More on Functions and Digraphs 109 A.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 A.2 Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 B More on the Principle of Mathematical Induction 113 C More on Equivalence Relations 118 1
Part I COURSE NOTES 2
Page 1 and 2: DISCRETE MATHEMATICS THROUGH GUIDED
Page 3: Contents I COURSE NOTES 2 1 Beginni
Page 7 and 8: to problems that are particularly i
Page 9 and 10: While working the next problems, be
Page 11 and 12: This idea of using functions to cou
Page 13 and 14: Figure 1.2: The union of four disjo
Page 15 and 16: The domain and co-domain of the fun
Page 17 and 18: •20. If B1, B2, . . . , Bm is a p
Page 19 and 20: In the last problem you used the fa
Page 21 and 22: ◦32. Prove that any function from
Page 23 and 24: nodes) and the line segments are ca
Page 25 and 26: niques which you have used and shou
Page 27 and 28: This illustrates one reason why som
Page 29 and 30: observation is the key to describin
Page 31 and 32: een used above is rather restrictiv
Page 33 and 34: to [10]. Give an inductive process
Page 35 and 36: 66. Are more moves required if the
Page 37 and 38: a recurrence can have infinitely ma
Page 39 and 40: prove your answer is correct, but s
Page 41 and 42: •86. In combinatorics, an (ordere
Page 43 and 44: ◦89. Now again suppose you are ch
Page 45 and 46: 3.2 Equivalence Classes Next is an
Page 47 and 48: are in different relative positions
Page 49 and 50: way in this course.) Sometimes �
Page 51 and 52: Calculation of Binomial Coefficient
Page 53 and 54: 118. Let S(k, n) denote the number
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Notice that a lattice path from (0,
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The suggestion intended by the hint
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Provided you can show that there is
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Chapter 4 Graph Theory 4.1 Undirect
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◦142. Find the degree of each ver
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A cycle in a graph is a walk which
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and then figure out in how many dif
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can begin to see this by working th
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185. Use the definition of the Rams
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(b) Draw another connected graph wi
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Figure 4.8: A graph. 1 2 4.6 Findin
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202. Use Stirling’s Formula to sh
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other restrictions are placed on th
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of x n is the number of ways to cho
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generating polynomial for the numbe
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Express this coefficient in the for
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chosen from the singleton set {i} .
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coefficient of x k in � ∞� ai
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In the last problem you represented
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The hardest part of generalizing yo
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248. A more elegant proof can be ob
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if and only if f(x) �= i for ever
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•262. Let G be a graph with verte
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Chapter 7 Distribution Problems 7.1
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◦271. In how many ways may you st
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•282. (a) Prove the Bell Numbers
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Index Kn, 20, 58 n!, Stirling’s f
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one-to-one function, 8 onto functio
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Part II REVIEW MATERIAL 108
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Exercise A.2. Here are some questio
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(a) Draw the digraph of the functio
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an n-element set has 2 n subsets. O
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of B1 is 2 k−1 . Consider the fun
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Appendix C More on Equivalence Rela
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DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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